Convergence \(S\)-compactifications (Q2922050)

Let \(X\) be a set, \(\mathcal{P}(X)\) be the power set of \(X\), \(\mathcal{F}\) be the set of all filters on \(X\) and for \(x\in X\) let \(\dot{x}\) be the ultrafilter on \(X\) generated by \(\{\{x\}\}\).NEWLINENEWLINEWe recall that:NEWLINENEWLINE(a) A pair \((X,q)\) (or just \(X\)) is called a \textit{convergence space} if \(X\) is a set and \(q:\mathcal{F}(X)\rightarrow \mathcal{P}(X)\) satisfies: (i) \(x\in q(\dot{x})\); (ii) \(\mathcal{G}\geq\mathcal{F}\) implies \(q(\mathcal{F})\subseteq q(\mathcal{G})\); and (iii) \(x\in q(\mathcal{F})\) implies \(x\in q(\mathcal{F}\cap \dot{x})\).NEWLINENEWLINE(b) A function \(f:X\rightarrow Y\) between two convergence spaces is \textit{continuous} if \(f^\rightarrow\mathcal{F}\rightarrow f(x)\) whenever \(\mathcal{F}\rightarrow x\).NEWLINENEWLINE(c) A triple \((S,\cdot,p)\) (or just \(S\)) is called a \textit{convergence monoid} if: (i) \((S,\cdot)\) is a commutative monoid with identity \(e\); (ii) \((S,p)\) is a convergence space; and (iii) the binary operation \((x,y)\mapsto x\cdot y\) of \(S\) is continuous.NEWLINENEWLINE(d) \(\lambda:X\times S\rightarrow X\), where \(X\) is a convergence space and \(S\) is a convergence monoid, is called a \textit{continuous action of \(S\) on \(X\)} if: (i) \(\lambda(x,e)=x\) for all \(x\in X\); (ii) \(\lambda(\lambda(x,s),t)=\lambda(x,s\cdot t)\) for all \(x\in X\) and all \(s,t\in S\); and (iii) \(\lambda\) is continuous.NEWLINENEWLINE(e) A triple \((X,S,\lambda)\) (or \((X,\lambda)\) or just \(X\)), where \(X\) is a convergence space, \(S\) is a convergence monoid and \(\lambda\) is a continuous action of \(S\) on \(X\) is called an \textit{\(S\)-space}.NEWLINENEWLINE(f) In a natural way concepts as \textit{Hausdorff}, \textit{regular}, \(T_3\), \textit{compactification}, etc. are defined for convergence spaces.NEWLINENEWLINE(g) An \(S\)-space \((Y,S,\mu)\) is an \(S\)-\textit{compactification} (\textit{regular \(S\)-compactification}) of an \(S\)-space \((X,S,\lambda)\) if: (i) \(Y\) is a compact Hausdorff (compact \(T_3\)) convergence space; and (ii) there exists a dense embedding \(f:X\rightarrow Y\) such that \(\mu\circ(f\times \text{id}_S)=f\circ \lambda\).NEWLINENEWLINEIn this paper, as the title suggests, the authors study \(S\)-compactifications of \(S\)-spaces. Here are some representative results: (i) In Theorem 2.3 (respectively, in Theorem 3.2), for some specific types of \(S\)-spaces, the authors give necessary and sufficient conditions for the existence of a smallest (regular) \(S\)-compactification; (ii) Example 2.6 witnesses that in general an \(S\)-space does not need to have a smallest or a largest \(S\)-compactification; (iii) Theorem 3.3 shows that if an \(S\)-space has a regular \(S\)-compactification then that space has a largest such compactification.